Question: If $60^a = 3$ and $60^b = 5,$ then find $12^{(1 - a - b)/(2(1 - b))}.$
Solution: We have that $a = \log_{60} 3$ and $b = \log_{60} 5,$ so
\[1 - a - b = \log_{60} 60 - \log_{60} 3 - \log_{60} 5 = \log_{60} \frac{60}{3 \cdot 5} = \log_{60} 4 = 2 \log_{60} 2\]and
\[2 (1 - b) = 2 (\log_{60} 60 - \log_{60} 5) = 2 \log_{60} 12,\]so
\[\frac{1 - a - b}{2(1 - b)} = \frac{2 \log_{60} 2}{2 \log_{60} 12} = \log_{12} 2.\]Therefore,
\[12^{(1 - a - b)/(2(1 - b))} = \boxed{2}.\]